Numerical Solution of a Nonlocal Fractional Boundary Value Problem By HDG Method

Öz

This paper is concerned with numerically solving of a nonlocal fractional boundary value prob-lem (NFBVP) by hybridizable discontinuous Galerkin method (HDG). The HDG methods have been successfully applied to ordinary or partial differential equations in an efficient way through a hybridization procedure. These methods reduce the globally coupled unknowns to approximations at the element boundaries. The stability parameter has to be suitably defined to guarantee the existence and uniqueness of the approximate solution. Some numerical examples are given to show the performance of the HDG method for NFBVP.

Anahtar Kelimeler

• Hybridizable discontinuous Galerkin methods,nonlocal fractional boundary value problem,stability parameter,hybridization

Kaynakça

1. [1] Geng, Fazhan, and Minggen Cui. “A reproducing kernel method for solving nonlocal fractional boundary value problems.” Applied Mathematics Letters 25, no.2 (2012): 818-823.
2. [2] Hilfer, Rudolf. Applications of fractional calculus in physics. ed. Singapore: World Sci-entific, 2000.
3. [3] Kilbas, Anatoly A., Hari M. Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006.
4. [4] Podlubny, Igor. Fractional differential equations. San Diego: Academic Press, 1999.
5. [5] Samko, Stefan G., Anatoly A. Kilbas, and Oleg I. Marichev. Fractional integrals and derivatives-Theory and Applications. Amsterdam: Gordon and Breach Science Publish-ers, 1993.
6. [6] Tenreiro Machado, J. A., Manuel F. Silva, Ramiro S. Barbosa, Isabel S. Jesus, Cecilia M. Reis, Maria G. Marcos, and Alexandra F. Galhano. “Some applications of fractional calculus in engineering.” Mathematical Problems in Engineering 2010 (2010): 34.
7. [7] Zhou, Yong, Jinrong Wang, and Lu Zhang. Basic theory of fractional differential equa-tions. Singapore: World Scientific, 2016.
8. [8] Ma, Ruyun. “Multiple positive solutions for nonlinear m-point boundary value prob-lems.” Applied Mathematics and Computation 148, no.1 (2004): 249-262.

9. [9] Ahmad, Bashir, and Juan J. Nieto. “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations.” Abstract and Ap-plied Analysis 2009 (2009):
10. [10] Khalil, Hammad, Rahmat Ali Khan, Dumitru Baleanu, and Samir H. Saker. “Approxi-mate solution of linear and nonlinear fractional differential equations under m-point lo-cal and nonlocal boundary conditions.” Advances in Difference Equations 2016, no.1 (2016): 1-28.
11. [11] Moshinsky, M. “Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas.” Bol Soc Mat Mexicana 7 (1950): 1–25.
12. [12] Nyamoradi, Nemat. “Existence of solutions for multi-point boundary value problems for fractional differential equations.” Arab Journal of Mathematical Sciences 18, no.2 (2012): 165-175.
13. [13] Shu, Xiao-Bao, and Qianqian Wang. “The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order . ” Computers & Mathematics with Applications 64, no.6 (2012): 2100-2110.
14. [14] ur Rehman, Mujeeb, and Rahmat Ali Khan. “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations.” Applied Mathematics Letters 23, no.9 (2010): 1038-1044.
15. [15] Zhong, Wenyong, and Wei Lin. “Nonlocal and multiple-point boundary value problem for fractional differential equations.” Computers & Mathematics with Applications 59, no.3 (2010): 1345-1351.
16. [16] Benchohra, M., S. Hamani, and S. K. Ntouyas. “Boundary value problems for differen-tial equations with fractional order and nonlocal conditions.” Nonlinear Analysis: Theo-ry, Methods & Applications 71, no.7 (2009): 2391-2396.
17. [17] Bai, Zhanbing. “On positive solutions of a nonlocal fractional boundary value problem.” Nonlinear Analysis: Theory, Methods & Applications 72, no.2 (2010): 916-924.
18. [18] El-Sayed, Ahmed M. A., and Ebtisam O. Bin-Taher. “Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order.” Electronic Journal of Differential Equations 2013, no.64 (2013): 1-8.
19. [19] Li, Xiuying, and Boying Wu. “Approximate analytical solutions of nonlocal fractional boundary value problems.” Applied Mathematical Modelling 39, no.5 (2015): 1717-1724.
20. [20] Celiker, Fatih, Bernardo Cockburn, and Ke Shi. “Hybridizable discontinuous Galerkin methods for Timoshenko beams.” Journal of Scientific Computing 44, no.1 (2010): 1-37.
21. [21] Cockburn, Bernardo, Bo Dong, Johnny Guzman, Marco Restelli, and Riccardo Sacco. “A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems.” SIAM Journal on Scientific Computing 31, no.5 (2009): 3827-3846.
22. [22] Cockburn, Bernardo, and Wujun Zhang. “A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems.” SIAM Journal on Scientific Computing 51, no.1 (2013): 676-693.
23. [23] Nguyen, Ngoc C., and Jaume Peraire. “Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics.” Journal of Computational Physics 231, no.18 (2012): 5955-5988.
24. [24] Cockburn, Bernardo, and Kassem Mustapha. “A hybridizable discontinuous Galerkin method for fractional diffusion problems.” Numerische Mathematik 130, no.2 (2015): 293-314.
25. [25] Mustapha Kassem, Maher Nour, and Bernardo Cockburn. “Convergence and supercon-vergence analyses of HDG methods for time fractional diffusion problems.” Advances in Computational Mathematics 42, no.2 (2016): 377-393.
26. [26] Karaaslan, Mehmet F., Fatih Celiker, and Muhammet Kurulay. “Approximate solution of the Bagley-Torvik equation by hybridizable discontinuous Galerkin methods.” Ap-plied Mathematics and Computation 285 (2016): 51-58.
27. [27] Karaaslan, Mehmet F., Fatih Celiker, and Muhammet Kurulay. “A hybridizable discon-tinuous Galerkin method for a class of fractional boundary value problems.” Journal of Computational and Applied Mathematics 333 (2017): 20-27
28. [28] Cockburn, Bernardo, Jayadeep Gopalakrishnan, and Raytcho Lazarov. “Unified hybrid-ization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems.” SIAM Journal on Numerical Analysis 47, no.2 (2009): 1319-1365.
29. [29] Rehman, Mujeeb, and Rahmat Ali Khan. “A numerical method for solving boundary value problems for fractional differential equations.” Applied Mathematical Modelling 36, no.3 (2012): 894-907.